/* MINDFIRE BLOG */

Introduction: Randomness, Belief, and Asteroids

Imagine flipping a coin or rolling a die—those feel truly random. Every flip and roll is fresh, unpredictable, and fair. When we say there is a 50% chance of getting heads or “1‑in‑6” odds of rolling a six, probability captures the genuine randomness of the experiment.

But what about something like Earth getting hit by a giant asteroid? That event doesn’t feel random in the same way—it’s a unique, one‑off cosmic occurrence based on deterministic laws. There is either an asteroid heading to earth that is going to hit us this year or there isn't. So where does probability come into play?

When we assign a probability to it, we’re really modeling our uncertainty—our lack of exact knowledge—not a repeatable random process like a coin flip.

For example, consider asteroids around 1 km in diameter—the kinds that could trigger widespread devastation or even global catastrophe. According to the Catalina Sky Survey, such an object strikes Earth on average once every 500,000 years (Catalina Sky Survey FAQ). That translates to an annual probability of approximately:

1 / 500,000 = 0.000002 (or 0.0002%)

In this case, probability is less about dice‑rolling-style pure randomness and more about representing our ignorance and expectations based on long‑term historical frequencies.

In the sections that follow, we’ll revisit classic probability concepts using coin flips and dice rolls—but we’ll also carry forward that perspective: probability is sometimes about real randomness, and sometimes about modeling how much we don’t know.

Variables and States

Let’s start with a simple example: flipping a fair coin. A coin can be in one of two states, either heads or tails.

We can represent this experiment using a random variable. Let’s call it c, for coin. The set of values that c can take is called its domain, and we write this as:

dom(c) = {heads, tails}

This means that the variable c can be in one of two states: heads or tails.

If the coin is fair, we assign equal probabilities to each outcome:

p(c = heads) = 0.5
p(c = tails) = 0.5

So when we say something like p(heads) = 0.5, we’re using shorthand for p(c = heads)— the probability that the coin lands on heads.

The important thing here is that we’re treating the coin as a variable (c), and the results it can take on (heads or tails) as states of that variable.

An event is an expression about a random variable such as two heads in a row or two heads out of three flips.

Summing Over Variables

Another important rule in probability is that the total probability over all possible states of a variable must add up to 1. This is called the normalisation condition. When you flip a coin you know that it must either land on heads or tails, so the total probability of both combined must add up to 1.0, meaning there is a 100% chance (certainty) it will be one of the two sides.

For our coin variable c, this means:

p(c = heads) + p(c = tails) = 1

Or, using more compact notation:

∑ p(c) = 1

This just says that when we consider all possible outcomes of the coin flip, one of them must happen. Either it lands heads, or it lands tails—there are no other options.

This idea of summing over all the states of a variable will show up again later when we start working with multiple variables.

In summary:

• A variable represents an uncertain quantity—like the result of a coin flip.
• The states of that variable are the specific outcomes it can take—heads or tails.
• The domain is the full set of those possible states.
• And the sum of the probabilities of all states of the variable must equal 1.

Interacting Variables: ANDs and ORs

Sometimes we’re interested in how two variables—or two events—relate to each other. One of the core rules in probability helps us figure out the chance of either event happening:

p(x or y) = p(x) + p(y) - p(x and y)

This accounts for the fact that if both events can happen together, we shouldn’t double-count that overlap.

Let’s look at an example with a 6-sided die.

• Let Event A be: the die shows an even number → {2, 4, 6}
• Let Event B be: the die shows a number greater than 3 → {4, 5, 6}

These events overlap at 4 and 6.

Now let’s calculate the probabilities:

• p(A) = 3/6 = 0.5 (since 2, 4, and 6 are even)
• p(B) = 3/6 = 0.5 (since 4, 5, and 6 are greater than 3)
• p(A and B) = 2/6 ≈ 0.333 (since 4 and 6 satisfy both)

Using the formula:

p(A or B) = p(A) + p(B) - p(A and B)
p(A or B) = 0.5 + 0.5 - 0.333 = 0.667

So there’s about a 66.7% chance that the die roll is either even, or greater than 3—or both.

This rule becomes especially useful when we work with more complex combinations of events later on.

Marginal Probability

Now that we've introduced how to handle two variables at once, it might be helpful to see how to take multiple variables and reduce back to one by summing the other variables. This is called calculating the marginal probability.

Let’s say we flip two coins. We’ll call them:

• c1: the result of the first flip
• c2: the result of the second flip

Each coin has two possible outcomes: heads or tails.

The full joint distribution (probabilities of combined events) includes all possible combinations:

• p(c1 = heads, c2 = heads) = 0.25
• p(c1 = heads, c2 = tails) = 0.25
• p(c1 = tails, c2 = heads) = 0.25
• p(c1 = tails, c2 = tails) = 0.25

These probabilities all add up to 1, as expected.

Now let’s say we only care about the result of the first coin. We can compute its marginal probability by summing over all outcomes of the second coin:

p(c1 = heads) = p(c1 = heads, c2 = heads) + p(c1 = heads, c2 = tails)
p(c1 = heads) = 0.25 + 0.25 = 0.5

p(c1 = tails) = p(c1 = tails, c2 = heads) + p(c1 = tails, c2 = tails)
p(c1 = tails) = 0.25 + 0.25 = 0.5

This process is called marginalisation. We’re taking a joint distribution of multiple variables and extracting just the part we care about—by summing over the rest.

Conditional Probability

Sometimes we don’t just want to know the probability of an event—we want to know the probability given that something else has already happened.

This brings us to conditional probability.

We write this as:

p(x | y) = p(x and y) / p(y)
or more formally:
p(x | y) = p(x, y) / p(y)

This definition tells us how to update our probability of x when we know that y is true. The numerator is the joint probability that both x and y happen, and the denominator scales it relative to how likely y is overall.

Let’s look at a simple dice example:

• Event A: the die shows a 6
• Event B: the die shows an even number (2, 4, or 6)

We want to know:

p(A | B) — the probability that the die shows a 6 given that it shows an even number.

Using the formula:

p(A | B) = p(A and B) / p(B)
p(A | B) = p(6) / p(even)
p(A | B) = (1/6) / (3/6) = 1/3

So if we already know that the die landed on an even number, there’s a 1 in 3 chance it was a 6.

This is the essence of conditional probability: we’re updating what we believe about one event in light of new information.

Bayes’ Rule: Deriving It Step by Step

Let’s start with something simple but powerful:

🔁 Joint probabilities are symmetric

The joint probability of two events x and y is the same regardless of their order:

p(x, y) = p(y, x)

This just says: the probability of both x and y happening is the same as the probability of both y and x happening. They describe the same outcome.

📘 Now recall the definition of conditional probability

By definition:

p(x | y) = p(x, y) / p(y)
This gives us the probability of x given that y has occurred.

Likewise:

p(y | x) = p(y, x) / p(x)
This gives the probability of y given that x has occurred.

Let's calculate p(x, y) now by multiplying both sides by p(y):

p(x,y) = p(x|y) p(y)

Note also that we could have instead done this as:

p(y, x) = p(y|x) p(x)

🔁 Substituting using the symmetry of joint probabilities

We know from earlier that:

p(x, y) = p(y, x)

So we can replace p(x, y) in the first equation with p(y | x) × p(x) (from the second equation above):

p(x | y) = p(x, y) / p(y)
p(x | y) = [p(y | x) × p(x)] / p(y)

✅ Bayes’ Rule

We’ve now arrived at the full expression:

p(x | y) = (p(y | x) × p(x)) / p(y)

This is Bayes’ Rule. It allows us to reverse the direction of a conditional probability—from p(y | x) to p(x | y)— by incorporating:

• p(x): our prior belief about x (i.e. the probability of x before we have any evidence)
• p(y): the overall likelihood of observing y
• p(y | x): how likely y is if x is true

Symptoms and Diseases: Using Bayes’ Rule to Flip the Question

Imagine you go to the doctor because you have a particular symptom—say, a persistent cough. Naturally, you want to know: What is the probability that you have a certain disease given this symptom?

This sounds straightforward, but here’s the catch: it’s often hard to directly measure how likely someone with that symptom actually has the disease. Symptoms can be caused by many things, and collecting data on all people with that symptom and their diagnoses is complicated.

However, it’s usually easier to measure two other things:

• How often people who have the disease exhibit that symptom. For example, suppose that 80% of people with disease X have a cough. This is the probability of the symptom given the disease:

p(cough | disease) = 0.8

• How common the disease is in the general population. For example, maybe disease X affects 1 in 1,000 people:

p(disease) = 0.001

With those two pieces of information, and using Bayes’ Rule, we can calculate what we really want:

What is the probability of having the disease given the symptom?

p(disease | cough) = [p(cough | disease) × p(disease)] / p(cough)

But what about p(cough), the overall probability of coughing? We can estimate that by considering:

• The chance of coughing if you have the disease: p(cough | disease) = 0.8
• The chance of coughing if you do not have the disease: say this is p(cough | no disease) = 0.1 (10% of people cough for other reasons)
• The overall chance of having the disease: p(disease) = 0.001
• The chance of not having the disease: p(no disease) = 1 - 0.001 = 0.999

To compute p(cough) — the overall probability that someone has a cough — we can break it down based on whether or not they have the disease. This follows the same logic we saw earlier with marginal probability, where we summed over the possible states of a variable.

In this case, we’re summing over two mutually exclusive groups: those with the disease and those without it. So we write:

p(cough) = p(cough, disease) + p(cough, no disease)

Using the definition of conditional probability

(p(x, y) = p(x | y) × p(y)),

we can rewrite this as:

p(cough) = p(cough | disease) × p(disease) + p(cough | no disease) × p(no disease)

This is also known as the law of total probability.

Substituting the values:

p(cough) = (0.8 × 0.001) + (0.1 × 0.999) = 0.0008 + 0.0999 = 0.1007

So even though the symptom is very common (10% of people get it regardless), the contribution from the rare disease (0.08%) is small in comparison.

Finally, apply Bayes’ Rule:

p(disease | cough) = (0.8 × 0.001) ÷ 0.1007 = 0.0008 ÷ 0.1007 ≈ 0.00795

This means:

Given that you have a cough, the chance that you have the disease is about 0.8% — or roughly 8 in 1,000.

Even though the disease is quite rare (0.1%), and the symptom common (coughing can be caused by many things), Bayes’ Rule helps us update the probability of disease in light of the symptom.

Conclusions

So what does this have to do with asteroids? I'm glad you asked.

Throughout this post, we've looked at how probability helps us reason under uncertainty:

• We defined variables and their possible states, like coin flips and dice rolls.
• We explored how probabilities add up across all possible outcomes of a variable.
• We looked at joint probabilities and how multiple variables interact.
• We introduced marginal probability by summing over hidden or unobserved variables.
• And we learned how Bayes’ Rule lets us flip the direction of a probability—from something easier to measure (like symptoms given disease) to something we really want to know (like disease given symptoms).

But all of this isn’t just about games of chance.

An asteroid hitting Earth isn’t like flipping a coin—it’s governed by physics. But we still use probability because we don’t know everything: we might not know the exact location of all near-Earth objects, or the precision of their orbits decades from now.

In this case, probability reflects uncertainty in our knowledge, not randomness in the universe. And yet, the same mathematical tools apply:

• We can define variables like “asteroid hits Earth this year” and assign probabilities to them.
• We can update those probabilities as we gather more evidence—say, spotting a new object in the sky.
• And we can reason about risk using exactly the same logic we used to diagnose a cough.

The rules of probability are universal—they help us model and update beliefs whether we're rolling dice, flipping coins, diagnosing disease, or watching the skies.